The Einstein-Maxwell Equations, Extremal Kähler Metrics, and Seiberg-Witten Theory

Nigel Hitchin has played a key role in the exploration of four-dimensional Riemannian geometry, and in particular has made foundational contributions to the theory of self-dual manifolds, four-dimensional Einstein manifolds, spinc structures, and Kähler geometry. In the process, he called attention to the profound mathematical interest of beautiful geometric problems that had previously only been considered by physicists. This chapter focuses on an interesting relationship between the four-dimensional Einstein-Maxwell equations and Kähler geometry, and points out some fascinating open problems that directly impinge on this relationship.